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Frobenius splitting and geometry of $$G$$-Schubert varieties. (English) Zbl 1160.14035
Let $$G$$ be a connected reductive group over an algebraically closed field $$k$$ of positive characteristic and $$B$$ be a Borel subgroup of $$G$$. Let $$X$$ be an equivariant embedding of the group $$G$$. A $$G$$-Schubert variety in $$X$$ is a subvariety of the form $$\text{diag}(G)\cdot V$$, where $$V$$ is a $$B\times B$$-orbit closure in $$X$$. If $$X$$ is the wonderful compactification of a semisimple group of adjoint type, then $$G$$-Schubert varieties coincide with the closures of G. Lusztig’s $$G$$-stable pieces [Parabolic character sheaves. I. Mosc. Math. J. 4, No. 1, 153–179 (2004; Zbl 1102.20030); II. Mosc. Math. J. 4, No. 4, 869–896 (2004; Zbl 1103.20041)].
The authors prove that $$X$$ admits a Frobenius splitting which is compatible with all $$G$$-Schubert varieties. Moreover, when $$X$$ is smooth, projective and toroidal, then any $$G$$-Schubert variety in $$X$$ admits a stable Frobenius splitting along an ample divisor. On the other hand, an example of a nonnormal $$G$$-Schubert variety in the wonderful compactification of a group of type $$G_2$$ is given. In the last section, a generalization of the Frobenius splitting results to the more general class of $$R$$-Schubert varieties is obtained.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds
##### Citations:
Zbl 1102.20030; Zbl 1103.20041
Full Text:
##### References:
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